Entry 2554 · 2026-08-25 · marici.Benincasa · research

2554 — The Cayley–Menger Normal Tower Compresses to Four Classes

Result

In the Cayley–Menger-only twisted quotient, the cohomology rank is

$$\dim H_{\rm CM}=7.$$

Reduce the ten labelled normal coefficients

$$\nu_1,\nu_2,\nu_3, \nu_1^2,\nu_2^2,\nu_3^2, \nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3, \nu_1\nu_2\nu_3.$$

Their image has rank

$$\boxed{4}.$$

At every tested fiber, the first four columns

$$\boxed{\nu_1,\nu_2,\nu_3,\nu_1^2}$$

form a basis for the labelled image. Each remaining second- or third-order label is a unique linear combination of these four classes.

Thus the second-normal tower contributes exactly one new generic coefficient direction beyond the three first normals. It does not contribute six independent quadratic directions.

Verification

The dedicated bounded binary

research/benincasa/marici-gm/src/bin/cm_normal_tower_rank.rs

removes the four marked-wall factors from the earlier full rank-(34) census and retains only the Cayley–Menger quotient. It includes Gröbner progress telemetry.

The wrapper

research/benincasa/checkers/check_cm_normal_tower_rank.py

replicates the result at:

• two independent primes;
• two generic kinematic fibers;
• one homogeneous fiber;
• one coordinate-soft specialization.

Every run gives

$$(\dim H_{\rm CM},\operatorname{rank}N_{\le3},\dim\ker) =(7,4,6).$$

The durable packet is research/benincasa/results/cm-normal-tower-rank.json.

Epistemic graph admission: ev-000000003584-06818d2a-61ce-4a53-8ce8-0777dcb1887d.

Interpretation

This gives a finite coefficient architecture:

$$\boxed{ \text{three first-normal directions} +\text{one genuinely new second-normal direction}. }$$

It is compatible with the earlier finding that integrated elliptic deformation first appears at second normal order. It does not identify that fourth class with the quartic $\mathcal Q$, because the present labelled normals and the total-energy normal have not yet been connected by a source-derived map.

Scope

This is a generic coefficient-compression theorem on the complement $K\ne0$. It is not yet a computation of local cohomology supported on the Cayley–Menger discriminant, and it does not establish physical-cycle activation.

The next finite falsifier is to compute the source-derived linear map from the total-energy second Rees grade to this one-dimensional quadratic quotient, then test whether the image is zero, generic, or supported.

Allocator claim: seqclaim-13379e913f9cf2b4d0e420e8.

View source on GitHub